Sato-Tate group of a modular form (reviewed)

The Sato-Tate group of a newform is a compact Lie group that one can attach to the Galois representation associated to the newform.

For newforms of weight $k=1$, the Sato-Tate group is simply the image of the corresponding 2-dimensional Artin representation, a finite subgroup of $\SL_2(\C)$.

For newforms of weight $k>1$ the Sato-Tate group is a subgroup of $\mathrm{U}(2)$ whose identity component is either $\mathrm{SU(2)}$ (for newforms without CM) or $\mathrm{U}(1)$ (for CM newforms) diagonally embedded in $\mathrm{U}(2)$.

The Sato-Tate conjecture implies that as $p\to\infty$ the limiting distribution of normalized Hecke eigenvalues $a_p/p^{(k-1)/2}$ converges to the trace distribution induced by the Haar measure of the Sato-Tate group.

The Sato-Tate conjecture for classical modular forms has been proved [MR:2630056, 10.4007/annals.2010.171.779].